A distributional version of functional equations and their stabilities

Generalizing the results in [J. Math. Anal. Appl. 286 (2003) 177–186; J. Math. Anal. Appl. 295 (2004) 107–114; Arch. Math., to appear; J. Math. Anal. Appl. 299 (2004) 578–586] that consider the Hyers–Ulam stability problems of several functional equations in the spaces of the Schwartz tempered distributions and the Fourier hyperfunctions we consider the stability problems of the functional equations in the space of distributions.     

Global attractivity results for nonlinear functional integral equations via a Krasnoselskii type fixed point theorem

In this paper, two results concerning the global attractivity and global asymptotic attractivity of the solutions for a nonlinear functional integral equation are proved via a variant of the Krasnoselskii fixed point theorem due to Dhage [B.C. Dhage, A fixed point theorem in Banach algebras with applications to functional integral equations, Kyungpook Math. J. 44 (2004) 145–155]. The investigations are placed in the Banach space of real functions defined, continuous and bounded on an unbounded interval. A couple of examples are indicated for demonstrating the natural realizations of the abstract results presented in the paper. Our results generalize the attractivity results of Banas and Rzepka [J. Banas, B. Rzepka, An application of measures of noncompactness in the study of asymptotic stability, Appl. Math. Lett. 16 (2003) 1–6] and Banas and Dhage [J. Banas, B.C. Dhage, Global asymptotic stability of solutions of a functional integral equations, Nonlinear Anal. (2007), doi:10.1016/j.na.2007.07.038], under weaker conditions with a different method.     

Logarithmically improved regularity criterion for the 3D generalized magneto-hydrodynamic equations

This article proves the logarithmically improved Serrin's criterion for solutions of the 3D generalized magneto-hydrodynamic equations in terms of the gradient of the velocity field, which can be regarded as improvement of results in [10] (Luo Y W. On the regularity of generalized MHD equations. J Math Anal Appl, 2010, 365: 806–808) and [18] (Zhang Z J. Remarks on the regularity criteria for generalized MHD equations. J Math Anal Appl, 2011, 375: 799–802).     

Generalized solutions of a nonlinear parabolic equation with generalized functions as initial data

In [H. Brézis, A. Friedman, Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pure Appl. (9) (1983) 73–97.] Brézis and Friedman prove that certain nonlinear parabolic equations, with the δ-measure as initial data, have no solution. However in [J.F. Colombeau, M. Langlais, Generalized solutions of nonlinear parabolic equations with distributions as initial conditions, J. Math. Anal. Appl (1990) 186–196.] Colombeau and Langlais prove that these equations have a unique solution even if the δ-measure is substituted by any Colombeau generalized function of compact support. Here we generalize Colombeau and Langlais’ result proving that we may take any generalized function as the initial data. Our approach relies on recent algebraic and topological developments of the theory of Colombeau generalized functions and results from [J. Aragona, Colombeau generalized functions on quasi-regular sets, Publ. Math. Debrecen (2006) 371–399.].     

Stability of functional equations of Drygas in the space of Schwartz distributions

In this paper we reformulate and prove the stability theorems of S.M. Jung and P.K. Sahoo [S.M. Jung, P.K. Sahoo, Stability of a functional equation of Drygas, Aequationes Math. 64 (2002) 263-273] in the spaces of generalized functions such as the Schwartz distributions and tempered distributions.     

Viscosity approximation methods for a family of finite nonexpansive mappings in Banach spaces

Viscosity approximation methods for a family of finite nonexpansive mappings are established in Banach spaces. The main theorems extend the main results of Moudafi [Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl. 241 (2000) 46–55] and Xu [Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004) 279–291] to the case of finite mappings. Our results also improve and unify the corresponding results of Bauschke [The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space, J. Math. Anal. Appl. 202 (1996) 150–159], Browder [Convergence of approximations to fixed points of nonexpansive mappings in Banach spaces, Archiv. Ration. Mech. Anal. 24 (1967) 82–90], Cho et al. [Some control conditions on iterative methods, Commun. Appl. Nonlinear Anal. 12 (2) (2005) 27–34], Ha and Jung [Strong convergence theorems for accretive operators in Banach spaces, J. Math. Anal. Appl. 147 (1990) 330–339], Halpern [Fixed points of nonexpansive maps, Bull. Amer. Math. Soc. 73 (1967) 957–961], Jung [Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 302 (2005) 509–520], Jung et al. [Iterative schemes with some control conditions for a family of finite nonexpansive mappings in Banach space, Fixed Point Theory Appl. 2005 (2) (2005) 125–135], Jung and Kim [Convergence of approximate sequences for compositions of nonexpansive mappings in Banach spaces, Bull. Korean Math. Soc. 34 (1) (1997) 93–102], Lions [Approximation de points fixes de contractions, C.R. Acad. Sci. Ser. A-B, Paris 284 (1977) 1357–1359], O’Hara et al. [Iterative approaches to finding nearest common fixed points of nonexpansive mappings in Hilbert spaces, Nonlinear Anal. 54 (2003) 1417–1426], Reich [Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 75 (1980) 287–292], Shioji and Takahashi [Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. Amer. Math. Soc. 125 (12) (1997) 3641–3645], Takahashi and Ueda [On Reich's strong convergence theorems for resolvents of accretive operators, J. Math. Anal. Appl. 104 (1984) 546–553], Wittmann [Approximation of fixed points of nonexpansive mappings, Arch. Math. 59 (1992) 486–491], Xu [Iterative algorithms for nonlinear operators, J. London Math. Soc. 66 (2) (2002) 240–256], and Zhou et al. [Strong convergence theorems on an iterative method for a family nonexpansive mappings in reflexive Banach spaces, Appl. Math. Comput., in press] among others.     

Existence and uniqueness results for impulsive functional differential equations with scalar multiple delay and infinite delay

In this paper, we discuss local and global existence and uniqueness results for first order impulsive functional differential equations with multiple delay. We shall rely on a nonlinear alternative of Leray–Schauder. For the global existence and uniqueness we apply a recent nonlinear alternative of Leray–Schauder type in Fréchet spaces, due to M. Frigon and A. Granas [Résultats de type Leray–Schauder pour des contractions sur des espaces de Fréchet, Ann. Sci. Math. Québec 22 (2) (1998) 161–168]. The goal of this paper is to extend the problems considered by A. Ouahab [Local and global existence and uniqueness results for impulsive differential equations with multiple delay, J. Math. Anal. Appl. 323 (2006) 456–472].     

On mean-square stability properties of a new adaptive stochastic Runge–Kutta method

We analyze the mean-square (MS) stability properties of a newly introduced adaptive time-stepping stochastic Runge–Kutta method which relies on two local error estimators based on drift and diffusion terms of the equation [A. Foroush Bastani, S.M. Hosseini, A new adaptive Runge–Kutta method for stochastic differential equations, J. Comput. Appl. Math. 206 (2007) 631–644]. In the same spirit as [H. Lamba, T. Seaman, Mean-square stability properties of an adaptive time-stepping SDE solver, J. Comput. Appl. Math. 194 (2006) 245–254] and with applying our adaptive scheme to a standard linear multiplicative noise test problem, we show that the MS stability region of the adaptive method strictly contains that of the underlying stochastic differential equation. Some numerical experiments confirms the validity of the theoretical results.     

An application of a global inversion theorem to an existence and uniqueness theorem for a class of nonlinear systems of differential equations

In this paper, a class of systems of nonlinear differential equations at resonance is considered. With the use of a global inversion theorem which is an extended form of a non-variational version of a max–min principle, we prove that this class of equations possesses a unique 2π$2\pi$-periodic solution under a rather weaker condition, for existence and uniqueness, than those given in papers [J. Chen, W. Li, Periodic solution for 2k$2k$th boundary value problem with resonance, J. Math. Anal. Appl. 314 (2006) 661–671; F. Cong, Periodic solutions for 2k$2k$th order ordinary differential equations with nonresonance, Nonlinear Anal. 32 (1998) 787–793; F. Cong, Periodic solutions for second order differential equations, Appl. Math. Lett. 18 (2005) 957–961; W. Li, Periodic solutions for 2k$2k$th order ordinary differential equations with resonance, J. Math. Anal. Appl. 259 (2001) 157–167; W. Li, H. Li, A min–max theorem and its applications to nonconservative systems, Int. J. Math. Math. Sci. 17 (2003) 1101–1110; W. Li, Z. Shen, A constructive proof of existence and uniqueness of 2π$2\pi$-periodic solution to Duffing equation, Nonlinear Anal. 42 (2000) 1209–1220]. This result extends the results known so far.     

A note on fixed point stability for generalized multivalued contractions

A classical lemma of Lim [T.C. Lim, Fixed point stability for set valued contractive mappings with applications to generalized differential equations, J. Math. Anal. Appl. 110 (1985) 436–441] has been extended in two ways, and one of them is used to obtain a stability result for generalized multivalued contractions.     

Remarks on differential Harnack inequalities

In this paper, we give a generalization of (global and local) differential Harnack inequalities for heat equations obtained by Li and Xu [J.F. Li, X.J. Xu, Differential Harnack inequalities on Riemannian manifolds I: linear heat equation, Adv. Math. 226 (5) (2011) 4456–4491] and Baudoin and Garofalo [F. Baudoin, N. Garofalo, Perelman’s entropy and doubling property on Riemannian manifolds, J. Geom. Anal. 21 (2011) 1119–1131]. From this we can derive new Harnack inequalities and new lower bounds for the associated heat kernel. Also we provide some new entropy formulas with monotonicity.     

On global asymptotic stability of nonlinear higher-order difference equations

In this paper, we generalize the main theorem of Liz and Ferreiro [E. Liz, J.B. Ferreiro, A note on the global stability of generalized difference equations, Appl. Math. Lett. 15 (2002) 655–659] and some other global stability results for nonautonomous higher-order difference equations to the case when contraction-type steps are incorporated together with the steps when the difference sequence can increase.     

Oscillation criteria for a forced mixed type Emden-Fowler equation with impulses

Some oscillation criteria for a forced mixed type Emden-Fowler equation with impulses are given. When the impulses are dropped, our results extend those of Sun and Meng [Y.G. Sun, F.W. Meng, Interval criteria for oscillation of second-order differential equations with mixed nonlinearities, Appl. Math. Comput. 15 (2008) 375-381], Sun and Wong [Y.G. Sun, J.S.W. Wong, Oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities, J. Math. Anal. Appl. 334 (2007) 549-560] for second-order forced ordinary differential equation with mixed nonlinearities, Nasr [A.H. Nasr, Sufficient conditions for the oscillation of forced superlinear second order differential equations with oscillatory potential, Proc. Am. Math. Soc. 126 (1998) 123-125], Yang [Q. Yang, Interval oscillation criteria for a forced second order nonlinear ordinary differential equations with oscillatory potential, Appl. Math. Comput. 135 (2003) 49-64] for forced superlinear Emden-Fowler equation, Kong [Q. Kong, Interval criteria for oscillation of second-order linear differential equations, J. Math. Anal. Appl. 229 (1999) 483-492] for unforced second order linear differential equations, and Wong [J.S.W. Wong, Oscillation criteria for a forced second order linear differential equation, J. Math. Anal. Appl. 231 (1999) 235-240] for forced second order linear differential equation.     

A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces

The purpose of this paper is to study the strong convergence of a general iterative scheme to find a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of variational inequality for a relaxed cocoercive mapping and the set of solutions of an equilibrium problem. Our results extend the recent results of Takahashi and Takahashi [S. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007) 506–515], Marino and Xu [G. Marino, H.K. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl. 318 (2006) 43–52], Combettes and Hirstoaga [P.L. Combettes, S.A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6 (2005) 486–491], Iiduka and Takahashi, [H. Iiduka, W. Takahashi, Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal. 61 (2005) 341–350] and many others.     

Initial value problems for nonlinear second order impulsive integro-differential equations of mixed type in Banach spaces

In this paper, through solving equations step by step, without any assumption of compactness-type conditions, we obtain unique solution of initial value problems of nonlinear second order impulsive integral-differential in Banach spaces. The results obtained generalize and improve the corresponding results of Guo and Chai in papers [D.J. Guo, Initial value problems for nonlinear second-order impulsive integro-differential equations in Banach spaces, J. Math. Anal. Appl. 200 (1996) 1–13; G.Q. Chai, Initial value problems for nonlinear second order impulsive integro-differential equations in Banach space, Acta Math. Sinica 20 (3) (2000) 351–359 (in Chinese)].     

Equivalent conditions for generalized contractions on (ordered) metric spaces

We establish a geometric lemma giving a list of equivalent conditions for some subsets of the plane. As its application, we get that various contractive conditions using the so-called altering distance functions coincide with classical ones. We consider several classes of mappings both on metric spaces and ordered metric spaces. In particular, we show that unexpectedly, some very recent fixed point theorems for generalized contractions on ordered metric spaces obtained by Harjani and Sadarangani [J. Harjani, K. Sadarangani, Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations, Nonlinear Anal. 72 (2010) 1188-1197], and Amini-Harandi and Emami [A. Amini-Harandi, H. Emami A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations, Nonlinear Anal. 72 (2010) 2238-2242] do follow from an earlier result of O’Regan and Petru?el [D. O’Regan and A. Petru?el, Fixed point theorems for generalized contractions in ordered metric spaces, J. Math. Anal. Appl. 341 (2008) 1241-1252].     

Maximum principles for a fourth order equation from thin plate theory

This paper focuses on a nonlinear equation from thin plate theory of the form Δ(D(x)Δw)−(1−ν)[D,w]+c(x)f(w)=0. We obtain maximum principles for certain functions defined on the solution of this equation using P-functions or auxiliary functions of the types used by Payne [L.E. Payne, Some remarks on maximum principles, J. Anal. Math. 30 (1976) 421-433] and Schaefer [P.W. Schaefer, Solution, gradient, and laplacian bounds in some nonlinear fourth order elliptic equations, SIAM J. Math. Anal. 18 (1987) 430-434].     

First-order hyperbolic pseudodifferential equations with generalized symbols

We consider the Cauchy problem for a hyperbolic pseudodifferential operator whose symbol is generalized, resembling a representative of a Colombeau generalized function. Such equations arise, for example, after a reduction-decoupling of second-order model systems of differential equations in seismology. We prove existence of a unique generalized solution under log-type growth conditions on the symbol, thereby extending known results for the case of differential operators [J. Math. Anal. Appl. 160 (1991) 93-106, J. Math. Anal. Appl. 142 (1989) 452-467].     

Decreasing solutions and convex solutions of the polynomial-like iterative equation

In this paper we prove the existence and uniqueness of decreasing solutions for the polynomial-like iterative equation so as to answer Problem 2 in [J. Zhang, L. Yang, W. Zhang, Some advances on functional equations, Adv. Math. (China) 24 (1995) 385-405] (or Problem 3 in [W. Zhang, J.A. Baker, Continuous solutions of a polynomial-like iterative equation with variable coefficients, Ann. Polon. Math. 73 (2000) 29-36]). Furthermore, we completely investigate increasing convex (or concave) solutions and decreasing convex (or concave) solutions of this equation so that the results obtained in [W. Zhang, K. Nikodem, B. Xu, Convex solutions of polynomial-like iterative equations, J. Math. Anal. Appl. 315 (2006) 29-40] are improved.